Assume \({\mathcal {K}}\)  is a finite set of finite algebras in a variety with a cube term.
Then it turns out that \(\operatorname{\text{\sc SIP}}({\mathcal {K}})\)  has a polynomial time reduction to \(\operatorname{\text{\sc SMP}}({\mathcal {K}})\) .
Indeed, given compact representations for two subalgebras \(\mathbf {B}\) , \(\mathbf {C}\)  of
\(\mathbf {A}_1\times \dots \times \mathbf {A}_n\)  with \(\mathbf {A}_1,\dots ,\mathbf {A}_n\in {\mathcal {K}}\) ,
Dalmau's algorithm [1]} yields a compact representation for \(\mathbf {B}\cap \mathbf {C}\)  in polynomial time.
Hence \(\operatorname{\text{\sc SIP}}({\mathcal {K}})\)  reduces to finding compact representations, which is polynomial time equivalent to
\(\operatorname{\text{\sc SMP}}({\mathcal {K}})\)  (Theorem REF ).
For the converse, we do not know whether \(\operatorname{\text{\sc SMP}}({\mathcal {K}})\)  reduces to \(\operatorname{\text{\sc SIP}}({\mathcal {K}})\)  in general.
However, it clearly does for idempotent algebras because for any elements \(a_1,\dots ,a_k,b\)  in
\(\mathbf {A}_1\times \dots \times \mathbf {A}_k\) , the subalgebra generated by
\(\lbrace a_1,\dots ,a_k\rbrace \)  contains \(b\)  if and only if that subalgebra has a nonempty intersection with \(\lbrace b\rbrace \) ,
the subalgebra generated by \(b\) . Hence for a finite set of finite algebras \({\mathcal {K}}\)  in an
idempotent variety with a cube term the problems
\(\operatorname{\text{\sc SMP}}({\mathcal {K}})\) , \(\operatorname{\text{\sc SIP}}({\mathcal {K}})\) , and finding compact representations
are all polynomial time reducible to each other.
